is really an inner product of a Jacobian matrix with a
quotient of two groups, K and L. (See Groupoid at Wikpedia)
A good starting point, from an economist point of view, is to describe the “value added chain”. Picking up a rock and painting a face and body is an example of a value added chain. Capital (rock and painting supplies), are combined by Labor (you), to produce My Pet Rock. One can describe the process as the following:
Labor (you), pick up Capital (rock).
Labor (you), wash Capital (rock).
Labor (you), paint Capital (rock).
Labor (you), use Capital (transport), My Pet Rock to market.
The above process is really hard to put into math terms. If one stares at the above process long enough one can get the illusion that some sort of pattern emerges. Labor (you), saw fit to pick a rock. Time is money, so Labor (you), added value. Capital (rock), had that shape that was of value. Value was added to the Capital (rock). Value is the common element. There is a branch of math called Category Theory which can help to place an order to the My Pet Rock process. I will not go deep into Category Theory (use your favorite search engine) (See figure 1’s).
- A category consists of a class of objects. Can be anything like A, B, etc…
- A class of morphisms designated by anything like f, g, h, etc… usually represented by arrows.
- Each morphism has an object as its domain and an object as its co-domain. Example f: A->B where A is the domain object of morphism f and B is the co-domain object. (fig 1a)
- Each object, example A, has an identity morphism which has the domain A and co-domain A. (IDa fig 1b)
- Each pair of morphisms, example f: A->B and g: B->C is a composite morphism g o f: A->C. The co-domain object of morphism f is the domain object of morphism g. Start from the right (f) and go left (g). (fig 1c)
A category has these rules:
- Each morphism f: A->B has identity composites (f o IDa = f) and (IDb o f = f)
- Each set of morphisms f: A->B, g: B->C and h: C->D is Associativity. (h o g) o f = h o (g o f) (fig 1d)
A simple example see fig 1e.
Check for Composition: f o h = IDb, f o g = IDb, h o f = IDa, g o f = Ida =>> OK
Check for Associativity: (h o f) o g = IDa o g = g
h o (f o g) = h o IDb = h hence h = g =>>> not OK
A good object domain for My Pet Rock process for labor’s transfer of value morphism is “labor” and the co-domain object is “labor-value”. The object domain for capital’s transfer of value morphism is “capital” and the co-domain object is “capital-value”. The perspective that “labor-value” object is a “capital” object and the reverse, “capital-value” object is a labor object is not a problem in Category Theory. Rock’s “labor” was in the form of catching your eye and you became the capital. My Pet Rock process is a composite morphism.
g:C->L and z:L->C are morphisms such that z o g = IDc and g o z = IDlSome notes:
- f o g = h o g does not imply g = h. It can be in some categories.
- There is only one ID morphism. A category having two morphisms starting and terminating from the same object indicates a “conversion”. Example: converting Fahrenheit to Celsius and the reverse conversion.
One can now translate the above My Pet Rock category into math using topology. We can think of the conversion of Labor into “labor-value” as a process that takes place in a time interval t. We have a family of morphisms f(L) (or f(C)) that takes the set Labor (or Capital) into the set “labor-value” (or “capital-value) within a time interval [0, 1]. I insist the topology conform to My Pet Rock category. This requirement means the topology space is simply connected. Making change or having the same number of laborers at equal cost are examples of a economic model being simply connected. It can be shown My Pet Rock is path connected and has a Trivial Fundamental Group. My Pet Rock process conforms to a Lie group. One can pick a representation of the Lie group. (See An Elementary Introduction to Groups and Representations as an example)
Output is a real number. I would like to pick GL(n, R). General Linear Group over R is a group of n x n invertible matrices with real entries. However, GL(n, R) is not a simply connected Lie Group. There is an out from the dilemma. GL(n, R) can be decomposed into its irreducible representation U(n). U (unitary) is a group of n x n matrix with complex entries whose column vectors are orthonormal. SU(n) is a unitary group whose determinant is one. SU(n) is a simply connected Lie Group.
SU(n) --> “real world” --> GL(n, R) process is a general scheme that helps economist analyze Neoclassical Theory. Economist believes there are NO infinities in Neoclassical Theory. This belief is born out of the real world. This belief is wrong. My Pet Rock category can have infinities. The infinities are the result of fraud.
Debt, Options, Futures, the Financial Bubble and Fraud
I can change My Pet Rock category into Bush's Crushing Debt category by changing the object “capital-value” into the object “debt-value”. Society is using FICO scores. Some firms are using FICO scores for their hiring decision. People change their behavior during times of economic stress. They borrow more in hopes they can pay off that debt when times are better. They become their own boss. Etc… Society hopes anyone can be their own boss, but Don Bush has changed that. One needs a prefect credit history, come from a rich family or knows someone who is rich. The capital markets have fled toward richer lands. FICO scores have now become a poverty trap policy.
I insist Don Bush’s policies conform to the Bush's Crushing Debt category. Don Bush’s policies extenuated the Financial Bubble. The Financial Bubble is a set pf accounting, regulation and policies that are outside the normal Neoclassical Theory, yet they interact. Don Bush’s meddling in the housing market through his proxy Fannie Mae and Freddie Mac is an example of the Financial Bubble affecting real economic growth.
I pick SL(n, C). Special Linear Group over C is a group of n x n invertible matrices with complex entries with a determinant of one. SL(n, C) is a simply connected Lie Group.
Here’s a good question: If Bernanke believes inflationary expectations are contained then why the financial bubble? See Chairman Ben S. Bernanke
At the Federal Reserve Bank of Boston’s 53rd Annual Economic Conference, Chatham, Massachusetts
June 9, 2008
Outstanding Issues in the Analysis of Inflation
Bernanke believes some inflation is needed for economic growth, yet the financial bubble has exceeded the needs of “US of A’s Real Economy”. What are the financial bubble’s inflationary expectations? I say the anchor is the $50+ trillion unfunded liabilities. Economist can use SL(n, C) as their math tool.
Viva La Revolucion!
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